I use Aran / worsted weight wool for a faster knit.
I begin with knitting my thumb first. I begin at the tip of the thumb and work until it is long enough to fit over my thumb. When my thumb is done, I put it onto waste yarn and put it aside.
Next, I begin my mitten at the finger-tips. I work increases steadily on each side (on both the front and back) until all my fingers can fit into the mitten. Then I work straight until the mitten fits onto my hand up to my thumb. Now it is time to put in the thumb.
I slip the first stitch from my thumb onto my needle and knit it together with the regular stitch on my needle. Then I do this a second time. Now I pass the first stitch on my right needle over the second stitch - this is basically a bind-off.
Once half the stitches from the thumb have been bound off, I thread my right needle through the remaining thumb stitches. If I have done it right, I will have the same amount of stitches on my needles as when I started.
here is a picture of my thumb already put into my mitten.